A Johnson-type bound is a positive number $J$ depending on the distance,
the blocklength and the cardinalities of the Alphabets constituting the
code. It garanties that a ``small" number of codewords are in any sphere
of radius $J$. By ``small" number, we mean a number of codewords which is
linear in the code blocklength and the cardinality of the code.
In our case, the Johnson-type bound for number
fields codes depends only on the code blocklength and its minimal
distance, and ``small" means polynomial in
$\sum_{i = 1}^n \log \Nm(\p_i)$.

The Johnson-type bound of~\cite[Section 7.6.1]{guruswami_phd} remains valid for number field codes.
For any prime ideal $\p\subset\OK$, the quotient $\OK/\p$ is a finite
field. Thus the $i$'th symbol of a codeword comes from an alphabet of size
$\Nm(\p_i) = |\OK/\p_i|$ and~\cite[Th. 7.10]{guruswami_phd} can be
applied. Let $t$ be the least positive integer such that
$
\prod_{i=1}^t \Nm(\p_i) > \left(\frac{2B}{d}\right)^d,
$ 
where $d = [K:\Q]$ and let
$
T = \prod_{i=1}^t \Nm(\p_i).
$  
Then, by \cite[Lem. 12]{guruswami_nb_fld}, the minimal hamming distance
of the number fields code is at least $n - t + 1$. Using~\cite[Th. 7.10]{guruswami_phd}, we can show that for a given message and $\varepsilon > 0$, only a ``small" number 
of codewords satisfy 
\begin{equation}
\label{eq:johnson_hamming}
\sum_{i = 1}^n a_i > \sqrt{(t + \varepsilon)n},
\end{equation}
where $a_i = 1$ if the codeword and the message agree at the $i$-th position, $a_i = 0$ otherwise. Thus, if our list decoding algorithm returns all the codewords having at most
$n - \sqrt{(t + \varepsilon)n}$ errors
then this number is garanteed to be ``small". Therefore, the Johnson bound appears to be a good objective for our algorithm. 
Note that we would derive a different bound by using weighted distances. In particular, by using the $\log$-weighted hamming distance
i.e. $d(x,y) = \displaystyle\sum_{i:x \neq y \mod \p_i} \log \Nm(\p_i)$,
the condition would be
$
\sum_{i = 1}^n a_i \log\Nm(\p_i) >
\sqrt{(\log T + \varepsilon ) \log N}
$.

